434
8. Nonrefleeting Boundary Conditions
i
(
!I
l .- -I :/
.........-.""/
/::;: I
.
I (c)
I
(d)
/ 1
- ..._ ----0 .7
x
1.0 0.7
x
1.0
FIOURE 8.7. Comparison ofnumerieal solutions to the advection equation obtained using
a Rayleigh damping wave absorber (dot-dashed eurve) or linear extrapolation (thin solid
eurve) and 6.x equal to (a) 1132, (b) 1164. or (e) 11256. In (d) the magnitude ofthe Rayleigh
damping eoefficient is doubled and 6.x = 1/64.
on the interval -1 :s: x :s: 1.125. Here N is the spatial index of the grid point on
the rigbt-boundary, and the Rayleigh damping coefficient is defined as
R()
{o.
ifx:s: 1;
x = a(1 + cos[8rr(x - 1)]) , otherwise,
so that tbe region 1 :s: x :s: 1.125 contains the wave absorber. Tbe Courant number
was one-half, U was I, and f/J was fixed at a constant value of zero at the right and
left boundaries. The initial condition was
1/!(X ,O)={O,
cos 2[rr(x - !)], otherwise.